Phonon Structure of Impurity-Related Optical Spectra in Insulators

The theory of optical processes associated with point imperfections in insulating crystals is briefly reviewed and a practical efficient computational procedure is developed for the detailed application of the theory to systems whose spectra exhibit marked vibronic structure. This procedure includes the following features: (a) an iterative scheme for extracting the effective one-phonon density of states from experimental data; (b) the convolution of the one-phonon spectrum to find the contributions of those $n$-phonon processes which yield discernible vibronic structure and the use of moment analysis for higher $n$-phonon processes; (c) inclusion of the lowest-order effects of quadratic coupling on the temperature dependence of the zero-phonon line's half-width and peak position; (d) a simple transformation between phonon operators in the ground and excited electronic states of the impurity which breaks the mirror symmetry between the absorption and emission spectra characteristic of the strict linear-coupling approximation. The absorption spectrum of the $N_1$ color center in NaCl, which exhibits a great deal of phonon structure, is used to illustrate certain aspects of the calculations. Good agreement between theory and experiment is obtained for this example.